Transition in Plane Parallel Shear Flows Heated Internally

1
Transition in Plane Parallel Shear Flows Heated
Internally

• Masato Nagata1 Sotos Generalis2
• 1Kyoto University, Japan
• 2Aston University, United Kingdom
• Email nagata_at_kuaero.kyoto-u.ac.jp

2
Overview

• Motivation
• Introduction
• Basic Equations
• Linear Stability
• Secondary Flow
• Stability of Secondary Flow Bifurcation points
  for tertiary flow
• Conclusions and Future work

3
Motivation

• Behaviour of the Earths mantle
• Nuclear Reactor Design Safety aspects
• Recent studies in various geometries and boundary
  arrangements in turbulent regime appropriate to
  nuclear reactor safety
• Examine bifurcation behaviour of strongly
  non-linear equilibrium solutions from laminar
  state to turbulent state Stability of Flow
• Employ recently developed techniques

4
Introduction
Geometrical configuration exhibiting the basic
flow profile with two inflection points in an
inclined layer internal heat source q.
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Introduction

• Apply Boussinesq approximation for fluid bounded
  between two parallel plates at TT0

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Introduction

• Non-dimensional description
• Length d
• Time d2/?
• Temperature qd2/2?Gr
• Physical properties are described by
  non-dimensional parameters
• Gr gßqd5/2??2 Grashof Number
• R Umaxd/ ? Reynolds Number
• Pr ?/ ? Prandtl Number
• Umax Maximum Laminar Velocity

7
Introduction
8
Introduction

• Basic Flow that satisfies fixed temperature
  T00 and no-slip u0 at z1

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Basic Equations

• Introduce velocity and temperature disturbances1
• 1M.Nagata S.Generalis,JHT,v124,p634 2002
• S.Generalis M.Nagata, JHT, in press 2003
• S.Generalis M.Nagata, Eurotherm 74, Heat
  Transfer in Unsteady Flows p1172003
• M. Nagata S.Generalis, Comptes Rendus Mecanique
  submitted 2003

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Basic Equations
Poloidal and toroidal parts of the velocity
fluctuations and temperature deviations
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Basic Equations
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Basic Equations
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Linear Stability (TW type disturbances)

• Ignore non-linear terms in fluctuations
• Ignore mean flow and temperature
• Identify maximum Grashof for laminar flow
• Travelling wave (TW) disturbances of the form
  (Squires theorem)

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Linear Stability (LR type disturbances)

• Ignore non-linear terms in fluctuations
• Ignore mean flow and temperature
• Identify maximum Grashof for laminar flow
• Longitudinal Roll (LR) disturbances of the
• form

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Linear Stability (TW and LR type disturbances)

• Critical Grashof number as a function of angle
  of inclination for LRs continous curve and TWs
  dash-dotted curves. Pr7,R0.

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Secondary Flow - LRs

• Newton-Raphson combined with Chebyshev
  collocation point method Pr7, R0

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Secondary Flow - LRs

• Total Mean Flow for R0, Pr7. B represents the
  basic flow contribution.

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Stability of Secondary Flow

• Infinitesimal 3-D disturbances are
  superimposed on 2-D equilibrium solution
  identify bifurcation for tertiary flow
• truncation level same as for steady solution
  

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Stability of Secondary Flow - LRs

• Instability boundaries of secondary LRs for
  Pr 0, R0. The dashed curve represents the
  linear neutral curve.

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Conclusions and Future Work

• Stability of incompressible flow in an inclined
  uniformly heated channel
• Linear various Pr, R stability infinitesimal
  disturbances Hopf bifurcation -streamwise -
  transverse TWs and spanwise LRs
• Secondary Flow - Stability of Secondary LRs
• General 3-D disturbance- Angle of Inclination
• Competing Instabilities
• Realistic case constant flux condition